1841. Hugo Luiz Mariano, João Schwarz First-order characterization of noncommutative birational equivalence
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Submission date: 7 September

Abstract:

Let Σ be a root system with Weyl group W. Let k be an
algebraically closed field of zero characteristic, and consider the
corresponding semisimple Lie algebra g_{k, Σ}. Then
there is a first-order sentence φ_Σ in the language
L=(1,0,+,*) of rings sucht that, for any algebraically closed field
k of char = 0, the validity of the Gelfand-Kirillov Conjecture for
g_{k, Σ} is equivalent to ACF_0 ⊢
φ_Σ. By the same method, we can show that the validity of
Noncommutative Noether's Problem for A_n(k)^W, k any
algebraically closed field of char = 0 is equivalent to ACF_0 ⊢
φ_W, φ_W a formula in the same language. As consequences, we obtain
results on the modular Gelfand-Kirillov Conjecture and we show that, for
𝔽 algebraically closed with characteristic p ≫ 0,
A_n(𝔽)^W is a case of positive solution of modular Noncommutative
Noether's Problem.